Math 361, Spring 2020, Assignment 3
From cartan.math.umb.edu
Revision as of 18:49, 14 February 2020 by Steven.Jackson (talk | contribs) (Created page with "__NOTOC__ ==Read:== # Section 19. ==Carefully define the following terms, then give one example and one non-example of each:== # Zero-divisor. # Integral domain. # Field....")
Read:[edit]
- Section 19.
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Zero-divisor.
- Integral domain.
- Field.
- Initial morphism (to a unital ring).
- Prime subring (of a unital ring).
Carefully state the following theorems (you do not need to prove them):[edit]
- Cancellation law (in an integral domain).
- Theorem relating unital subrings of fields to integral domains.
- Fundamental Theorem of (Ring) Homomorphisms.
Solve the following problems:[edit]
- Section 19, problems 1, 2, 3, 6, 7, 8, and 9. (Note: the characteristic of a unital ring is the non-negative generator of the kernel of the initial morphism. So for problems 6-9, begin by trying to understand the initial morphism and computing its kernel. We will have much more to say about the characteristic of a ring next week.)
- Describe the prime subring of M2(Z3). (Hint: first make a table of values for the initial morphism.) Is the prime subring all of M2(Z3)?
- Describe the prime subring of Z3×Z3. Is it all of Z3×Z3?
- Describe the prime subring of Z2×Z3. Is it all of Z2×Z3?
- Prove that Z2×Z3 is isomorphic to Z6. (Hint: use the Fundamental Theorem.)
- Prove that Z3×Z3 is not isomorphic to Z9. (Hint: compute the characteristic of each ring.)